Laminar counterflow spray diffusion flames: A comparison between experimental results and complex chemistry calculations

COMBUSTION A N D F LA ME 95:261-275 (1993)

261

Laminar Counterflow Spray Diffusion Flames: A Comparison Between Experimental Results and Complex Chemistry Calculations N. DARABIHA, F. LACAS, J. C. ROLON and S. CANDEL Laboratoire EM2C, CNRS, Ecole Centrale Paris, 92295 Chatenay-Malabry Cedex, France Experimental and numerical studies of laminar flames formed by the counterflow of a monodisperse fuel spray with an air stream are reported in this article. In this simple configuration it is possible to analyze the influence of the phase transfer terms on the flame structure. The experimental setup used to produce such laminar spray diffusion flames is first described. A set of experiments are carried with liquid heptane fuel sprays. The flame is characterized with a laser sheet imaging system and with a particle sizing apparatus based on laser light diffraction. Results of a numerical study are then presented. The two phase-reacting flow equations are solved through Newton iterations and adaptative gridding using detailed transport and complex chemistry. An iterative procedure is devised to solve the gas- and liquid-phase balance equations. Comparison between experimental and numerical values of the diameter are found to be in good agreement.

INTRODUCTION In many practicle combustion systems, the fuel is injected as a liquid phase. This is the case for example in Diesel engines or in turbojets, where the fuels are liquid hydrocarbons. Similar physical processes are found in rocket motors. Although one of the propellants may be injected in gaseous form (e.g., hydrogen), the oxidizer is usually liquid (e.g., LOX). To design these combustors and predict their performance with a reasonable accuracy, it is necessary to have an understanding of the flame structure and multiphase dynamics of these reacting flows. This explains the large number of studies that have concerned problems of injection, atomization, and spray combustion, both experimentally and numerically. Comprehensive reviews of many aspects of two-phase combustion are due to Williams [1] and Faeth [2]. Some of the basic theory may be found in Refs. [3-5]. Many articles in this area deal with the case of a single droplet surrounded by a diffusion flame in an oxidizing gaseous atmosphere. Theoretical and empirical laws have been derived in this case for a wide range of problems including situations where the droplets are moving with respect to the ambient gas. Fewer studies are concerned with the socalled group combustion case, where the Copyright © 1993 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.

droplet density is sufficiently large and interactions between droplets become important. Typical studies belonging to this class are contained in Refs. 6 and 7. For dense droplet clouds, the interaction between the flame front and the droplet group becomes the leading phenomenon. The study of propagating flame fronts in droplet clouds convected in a gaseous stream poses difficult experimental problems. Because a propagating flame is not easily stabilized in a channel, the experimental time duration is short and one finds a certain variability in measurements carried under identical conditions. When the flame progresses in a channel filled with a spray it is also difficult to obtain quantitative data. These problems may be overcome by using a counterflow geometry (Fig. 1). In this configuration, the flame front is fixed in space allowing detailed measurements of the combustion zone. The counterflow system has been commonly used in the case of gaseous reactants, [8-11]. While it seems natural to study the same flow geometries in the case of reactive sprays, only a small number of studies deal with this problem. This is rather surprising since the counterflow geometry is well suited to systematic studies of the effect of strain on flames formed by spray. Theoretical work is reported in Refs. 12 and 13 and some 0010-2180/93/$6.00

262

N. DARABIHA ET AL. Upper burner

with this method compare favorably with the measurements. EXPERIMENTAL SETUP

~lion point

Ion

Fig. 1. Spraycounterflowdiffusionflame configuration. interesting experiments were conducted in the case of solid coal particles carried in a gas flow [14, 15]. The combustion of a fuel spray near a stagnation point formed by a jet impinging on a flat plate is investigated in Ref. 16 in the premixed situation where the droplets are transported by a coflowing mixture of methane and air. The experiment is complicated because a polydisperse spray is formed and it is then necessary to follow different classes of droplet diameters. This article describes experiments and calculations of laminar flames established by the counterflow of a monodisperse fuel spray with a stream of air at atmospheric pressure. Special attention is given to the atomization system, which determines the quality of the experiment. We report initial results, both qualitative and quantitative, obtained with a spray of n-heptane. Experimental measurements are based on laser sheet tomography combined with image processing and on particle sizing by laser light diffraction. A numerical investigation of the problem is then presented. The flame is modeled with the low Mach number equations for the gaseous and liquid phases. A similarity solution may be derived under conditions that are not too restrictive. This solution is then determined by employing an iterative procedure which involves successive solution of the gas and liquid phase balance equations using detailed transport and complex chemistry. Results obtained

The production of counterflow laminar flames is now well documented. Tsuji [8] provides a broad review of the work carried on this subject and describes a number of devices that may be used to produce such flow geometries. Among the various arrangements the counterflow burner is chosen to avoid interactions between the droplet cloud and solid boundaries which could lead to droplet collection and liquid film formation on the stagnation plate. The heat losses to the boundaries are also minimized. The good optical access of the counterflow geometry is another reason for our choice. Finally, the main advantage of the counterflow geometry is that it may be used in the premixed and nonpremixed modes. Because the nonpremixed spray flame is specifically interesting for diesel and rocket engine combustion, it is studied in the present article. The high pressures that exist in these applications are not reproduced in the present experiment. The burner operates at atmospheric pressure in order to reduce the complexity of the setup. The experimental system comprises two axisymmetric nozzles fixed on a vertical structure. Each burner features two concentric jets. The internal jet conveys the reactant while the surrounding nitrogen jet acts as an aerodynamic screen preventing the formation of a diffusion flame between the reactants and the ambient air. The diameters of the nozzle are 20 mm (internal jet) and 40 mm (external jet), respectively. The distance h between the two burners may be varied but will be kept fixed in the present experiment ( h - - 2 0 mm). The lower burner may be moved in the xy plane (Fig. 2) in order to align the nozzle central axis. The two burners, including the berth, may be displaced in the x and y directions to explore the whole flame zone without moving the optics or diagnostics. The translations are computer controlled with two steping motors. This design is well suited to laser diagnostic measurements because the optical instrumentation is held fixed on a modular structure surrounding the

LAMINAR COUNTERFLOW SPRAY DIFFUSION FLAMES

~

honeycomb ~

n -

Heptane

~

Flame ~

N2shield atomizer Carriergas

Fig. 2. Experimental setup used to produce the counterflow laminar diffusion flame with fuel spray. The lower burner is shown as a cut.

combustor. This structure is placed on a heavy steel table fastened to a concrete block thus limiting the vibration level. The combustor design is also fully independent from the gas flow control system. Different reactant supply configurations may be used allowing both premixed and nonpremixed flame operations. The upper and lower burners are fed with separate lines. For the most general use, each burner is supplied with nitrogen, oxygen, and a fuel gas all stored in 200 bar reservoirs. A nitrogen flow creates an aerodynamic screen around each jet. Flow rates are measured by sonic nozzles and a large range of operating conditions may be covered by changing the sonic nozzle diameters. The flow control system comprises eight pressure measurement points which deliver pressure signals to a computer through an AD converter. The gas mixture is prepared in two separate chambers located on the downstream side of the sonic nozzles. Figure 2 shows a schematic view of the supply system. More details on components used in the setup are given in Refs. 17 and 18. The spray preparation is a critical problem in reactive spray experiments. Interpretation of the experimental data is greatly simplified if certain requirements are precisely fulfilled. The liquid flow rate should be sufficiently large but should not depend on the gaseous flow rates. It is desirable to have a narrow distribution of droplet sizes and the spray velocity should be initially low. Mechanical or nozzle-like atomizers yield large flow rates, but the spray distribution is broad and the initial velocity of the spray is large. Systems based on pulsed jet instabilities have very good spray characteris-

263

tics, but yield small liquid flow rates. Under these circumstances the ultrasonic atomizer appears as the best choice. This device uses the dynamic properties of a free liquid surface submitted to a vibration at ultrasonic frequencies [19-21] to produce a homogenous cloud of droplets. If the power (i.e., the amplitude) of the oscillations is sufficient for a given frequency, droplets are formed over the surface. The droplet size may be changed by selecting a different excitation frequency. The system comprises three different components: the transducer, the electrical power supply and the liquid flow control system. The transducer is driven by a piezoelectric actuator. The design of such piezoelectric transducers is well documented (see Refs. 22 and 23, among others). The device comprises ceramic rings (PbTiO 3) placed in sandwich between two metallic plates. A view of the atomizer is presented in Fig. 3. One of the metallic parts has a variable section and acts as a vibration amplifier. The atomizer also comprises a liquid supply system designed to handle different liquid fuels such as ethanol and heptane. Fuel is stored in a pressurized tank (nitrogen for safety reasons). The liquid flow rate is measured by an electronic flowmeter to allow real-time computer control. According to Mizutani [21] the relation between the frequency and the droplet size has the following form: 43 = 0.149(8,n-~qq ~ ) , where dl, o-, and pq are, respectively, the droplet diameter, the surface tension, and the Amplifier

I IOVAC °'

•

I *'1SOY=:

I

Piezoelectric ceramics

j_ ~

Liquid supply

Fig. 3. Schematic view of the atomization device including atomizer, electric power, and liquid supply.

264 specific mass of the liquid and f is the excitation frequency (all in CGS units). A mean droplet diameter of the order of 20 /zm is obtained with a frequency of 50 kHz. The peak frequency may be easily changed by modifying the dimensions of the metallic parts. The transducer is excited by a + 150-V, 50kHz continuous signal delivered by an amplifier operating with a high voltage dc power supply. A square wave is used to maximize the energy transmitted to the ultrasonic transducer. Measurements of droplet size reported in Refs. 19 and 21 for this kind of system indicate a good quality of the spray distribution. The maximum flow capacity of the atomizer system is about 0.15 cm 3 s -]. DIAGNOSTIC TECHNIQUES Figure 4 provides a general view of the tomographic set up used to visualize the spray. A 15-mW helium-neon laser constitutes the light source. The light beam originating from the laser is expanded through two cylindrical lenses (focal length 6.35 and 150 ram, respectively) to produce a sheet having 1 mm in thickness and a 23 mm width. The light scattered by the fuel droplets is collected with a CCD camera. The image may then be processed to obtain more quantitative data on the flame structure. If the spray distribution is monodisperse in each section, the amplitude of the scattered light measured along the vertical axis will give a good representation of the droplet diameter. To get this information the image recorded by the camera is transferred to a Mac II computer for further processing. A set of frames is averaged and the background light is substracted. This

N. DARABIHA ET AL. yields the scattered light intensity on the axis and the light emitted by the flame itself. It is then possible to extract the evolution of the light intensity along the axis for further comparisons with the droplet diameter evolution deduced from the calculations. It is also possible to measure the distance between the vaporization front and the flame front and compare this quantity with that deduced from calculations. To obtain a preliminary quantitative measurement of the droplet size distribution along the y axis we use a laser light diffraction technique. A general view of this setup is given in Fig. 5. When a cloud of spherical particles is illuminated by a monochromatic parallel coherent beam, a diffraction diagram is formed. This diagram is a sum of elementary diffraction patterns weighted by the particle number in each class of sizes. Inversion of these data provides the droplet size distribution. The light source is a 5-roW helium-neon laser (632.8 nm). A linear wedge is used to trim the laser beam to a diameter of 1 mm. This diameter does not enable a spatial resolution smaller than 1 mm. But, for a smaller diameter, the signal-to-noise ratio is too small to allow the measurements. This is a major limitation of the laser light diffraction method. The parallel light beam is diffracted by the spray and collected with a lens (300 mm focal length and 150 mm diameter) and a CCD 1024 pixel linear array (Thomson TH7808). The pattern formed on the photodiode array is collected and averaged on a digital oscilloscope and transfered to a computer. The scattered light is the sum of the individual patterns corresponding to the droplet classes that scatter the laser beam during each exposure. The pattern obtained by

I OIclIIoICOpO

I,"--~

o o I

Llnlmr LL I I l l - . ~ /

Fig. 4. Schematic view of the experimental setup used for the laser sheet visualization of the flame.

~

// )

~

"-----~j/

I'--~

Llnl~r w,,~p

Oropa*tcloud

Fig. 5. Schematic view of the experimental setup used for the light diffraction sizing technique.

LAMINAR COUNTERFLOW SPRAY DIFFUSION FLAMES averaging 50 exposures is then numerically inverted to obtain the droplet size distribution and the mean droplet diameter. More details on the set up and on the algorithms may be found in Refs. 24 and 25. EXPERIMENTAL RESULTS A direct picture of the flame and of the laser light scattered by the spray is given in Fig. 6. The maximum of the light intensity is detected in the flame and is due to the spontaneous emission. It may clearly be seen that the intensity of the light scattered by the droplets rapidly decreases short before the flame front. This sharp decrease of the scattered light does not mean that the droplets have completeIy disappeared, but merely that their diameter has diminished in a very fast way. This measurement indicates that a vaporization front is formed in the near vicinity of the flame front. In this vaporization front, the spatial gradient of the droplet diameter is quite sharp. For small droplets (i.e., around 20/zm) the characteristic time of vaporization is of the same order as the typical time of the combustion process.

265

An average of several frames followed by some pixel filtering provides the relative intensity of the light emitted by the flame or scattered by the droplets along the axis of the device. Figure 7 displays a typical plot of the intensity as a function of the vertical coordinate along y axis. The liquid mass flow rate is in this case 0.033 g s i (mass flux is equal to 0.0105 g cm -2 s-1), the diluted propane mass fraction is YC3H8 = 0.16 and its total (C3H 8 + N 2) mass flow is 0.315 g s i (mass flux is equal to 0.10 g cm -2 s-1) and the mass flow rate of air delivered by the other nozzle is 0.28 g s-1 (mass flux is equal to 0.089 g c m -2 S - 1 ) . Liquid film formation was not observed in typical experiments. The left of the curve corresponds to the lower burner, where the droplets are injected in the flow. The right side corresponds to the upper burner, where the air stream is injected. From left to right, one may note that the light scattered by the droplets slowly increases reaches a maximum and then decreases at a fast rate in the vaporization front. One then observes the light emitted by the flame which reaches a maximum and then decreases. Four separate regions may be distinguished in this diagram.

Fig. 6. Direct view of the spray eounterflow diffusion flame. The droplets are visualized by a helium-neon laser sheet. The droplet cloud is injected by the lower nozzle.

266

N. DARABIHA ET AL.

|

_,

I

O,

'111 I

I I

i

i i i i [ i i ill 1, 1

I Ii

i i I i i i i ] i i 2,

ylonOtll ( crn)

Fig. 7. Intensity of the light scattered by the droplets or emitted by the flame on the burner axis and collected on the CCD camera.

The first region corresponds to a slow vaporization of the droplets. The weak increase of the light intensity cannot be due to condensation, because no gaseous n-heptane is present in the cold flow. It is most probably due to the nonconstant intensity of the laser light: the expanded laser beam retains a gaussian energy profile. As a consequence the light scattered by the droplets also shows an increasing profile from the side to the center of the laser sheet. The second region corresponds to the very fast decrease in the scattered light. As it may be seen in the global picture, the gradients in this area are very sharp. The thickness of this zone is less than 1 mm, which indicates a very fast vaporization of the 20-/zm-diameter droplets. The light intensity after this region is very low and close to the level of the ambient light background, showing that in this area the droplet diameter is very small if there are still any droplets. In the third and fourth regions the light intensity increases again, but the source of radiation is now the light emitted by the flame. This light reaches a maximum and then decreases again which is consistent with typical distribution of the heat release rate in a laminar counterflow diffusion flame. The slower decrease on the cold oxidizer side is due to the fact that the flame is slightly curved. The light emitted by the curved skirts of the flame and collected by the CCD camera interferes with the light emitted on the axis. The axial profile of intensity yields estimates of the distance separating the flame front and

the vaporization front. This distance is typically of the order of 2 mm but its exact value depends on the conventions used to define the flame front and the vaporization front locations. A more quantitative measurement of the droplet size distribution in the first region is obtained with the light diffraction technique described above. The burner is moved vertically by steps of 1 mm. At each step, the droplet cloud is analyzed and the diffraction pattern of the spray is stored. After storing all the patterns at different heights, we carry numerical inversion to obtain the droplet size distribution of the cloud and deduce the mean droplet diameter. The spatial resolution of the method determined by the diameter of the laser beam is 0.8 mm while the resolution in droplet diameter is 1 /~m. This uncertainty is fixed by the optics and the inversion algorithm. Figure 8 shows a plot of the Sauter mean droplet diameter as a function of the y coordinate. The experimental conditions are as before. Two different regions may be distinguished in this plot. In the first region, the mean droplet diameters is slowly diminishing. The vaporization of the droplets is slow in this area. The staircase aspect of the measurements is due to the resolution in size of the measuring system. In the second region, the droplets seem to vaporize rapidly over a distance which extends over less than 1 mm (equal to the spatial resolution of the measuring system). This curve is in good agreement with the shape observed with the laser sheet visualiza24

•

i

,

i

20 16 12 8 4 X.X

0 0

4

8

"r~

12

16

20

y (ram) Fig. 8. Mean droplet radius, measured with the particle sizing technique, plotted as a function of the axial coordinate y.

LAMINAR C O U N T E R F L O W SPRAY DIFFUSION FLAMES tion technique. In both experiments, one observes a slow evolution of the spray, due to "natural" vaporization of the liquid fuel followed by a very fast vaporization step, due to the heating of the flame. It may also be noted that the mean diameter of the droplets delivered by the injector is 22 /xm, which is quite close to the theoretical 20 /zm expected from the atomization device.

267

x momentum

dug

pgUg 2 -~ pgUg dy

-dy + J + ntht(U t -

(2)

Ug) - n F x.

Energy

GOVERNING EQUATIONS

d

Consider a counterflow of oxidizer and diluted fuel spray as shown in Fig. 1. To describe this configuration one may use a low-Mach number approximation with full coupling between liquid and gaseous phases. If the spray is not too dense, a spherically symmetric single-component droplet surrounded by a quasi-steady spherically symmetric film model may be used to obtain the rate of droplet vaporization. The gas phase is a mixture of K species. The specific index r is assigned to the component which exists in both phases (the constituent of the liquid phase). As shown in Ref. 13, the problem may be studied with a similarity analysis of both phases. This similarity analysis leads to solutions of the form: pg = & ( y ) , Ug = xU~(y), Vg = vg(y), Tg = Tg(y), Yk = Yk(Y), k = 1 , . . . , K , Pl = Pt(Y), ut = xUI(Y), vl = vl(Y) and T/ = Tl(y), where x and y, respectively, denote radial and axial coordinates, subscripts g and l, respectively, designate the gaseous and liquid phases, p, u, v, T, and Yk, respectively, denote density, radial velocity, axial velocity, temperature, and the kth species mass fraction. Ug and Ul describe the y dependence of the transverse velocities Ug and u t, respectively. Assuming a constant radial pressure-gradient d p / d x = J along the axial coordinate [26, 27], the system of equations that describes the two-phase reactive flow may be cast in the following form (see Ref. 13 and the appendix for more details):

dy(

&VgCpgdy

-

~_~ PgYkVkyCpgk

dy

k= 1 K

-- E (hkWkO)k) k=l + nfntCt, g~(T1 - Te) - nrhtq.

(3) Species d~ pgUg dy

d dy

-

+ ¢3krnth

I -

n t h t Y k,

k = 1,-.., K.

(4) Liquid phase Mass

dm I Ul dy

rh I .

(5)

x momentum

dg

mtUt 2 + mtvl---~-y = F x .

(6)

y momentum dvt mtvt--~y = m t g + fv.

(7)

Gas phase Mass

(l + j)pgUg +

Energy d pg Vg dy

aT,

nrht"

(1)

mlCplUl--~- = rhl( q - L).

(8)

268

N. D A R A B I H A E T A L .

Droplet number density dnu l

(1 + j ) n U t + - -

dy

= 0.

(9)

In Eqs. 1-9 the parameter j is equal to 0 or 1 for planar or axisymmetric configurations, respectively, and ~kr is equal tO 1 for the species which is present in both phases (Skr = 1 when k = r) and 0 for all other species. In these equations n stands for the droplet number density, rh t is the mass vaporization rate of a single droplet, and /Xg denotes the local viscosity of the gaseous mixture. Standard notations are used for the other quantities; Cpg is the local constant pressure heat capacity of the gaseous mixture, Cpgk is the constant pressure heat capacity of the kth species, Cpgr is the constant pressure heat capacity of the species existing in both phases, Vky i s the diffusion velocity of the kth species in the y direction, h k is the specific enthalpy of the kth species, Wk is the molecular weight of the kth-species, o% designates the molar rate of production of the kth species, q represents the heat transfered to the droplet interior, m r is the droplet mass, fx and fy are the radial and axial components of the drag f, Fx is the reduced radial component of the drag (Fx --fx/X), and L is the latent heat of vaporization. Equations 5-9 are derived from balance equations in liquid phase and the derivation is given in the appendix. The system of equations is completed by specifying the equation d J / d y = 0 indicating that the radial pressure gradient is constant in the axial direction [26, 27] and the equation 1 3 m r =-grrdlPq, where d r denotes the droplet diameter and pq the specific mass of the liquid. Note that the liquid phase density Pr is related to m r by the equation Pr = nmr. If the droplet diameter d r and droplet density n are small enough, one may consider that the liquid phase volume is negligible when compared with the gaseous phase volume. In that case the system of equations is completed by specifying the ideal-gas equation for the gas phase giving pg as a function of Tg and p (pg = p W / R T g ) . The transport coefficients, thermodynamic properties and chemical production rates are calculated in terms of the state variables and

their gradients with standard expressions involving detailed transport and complex chemical kinetics [28, 29]. Droplet interactions should be taken into account in dense sprays. These interactions are weak if the mean droplet separation exceeds ten diameters and one may then specify mass vaporization rate of each droplet rh r in terms of the single droplet expression [30, 31]: rh r = 2rrdtpgDgfln(1 + BM),

where Dgf is the diffusivity of the species being vaporized and BM is the transfer number, B M = (Y~s - Y~)/(1 - Yrs)" In this expression Yr is the mass fraction of the species that vaporizes and Yrs represents the saturation mass fraction at the droplet surface given by

WklX. Yrs =

WkrXrs + (1

where W = (E~=1, i . , . W i X i ) / ( 1 - X r ) . The saturation mole fraction Xrs may be expressed as a pressure ratio Xr~ = P r J P , where P,-s is the vapor pressure at the temperature T r [32]. It will be assumed that the droplets have a spatially uniform temperature. In fact, the characteristic time of heat transfer in the droplet is given by t*

Prdl2cM

4& For liquid heptane droplets having a diameter of 20 tzm, this time is about 10 ms which is one order of magnitude smaller than the characteristic time of the flow. The heat transferred from the gaseous phase to each droplet q is modeled according to [5] by q = Cpgi(Tg - T , ) / B T ,

where B r =- e rhl/2~pgfDgfdl - - 1. The subscript f denotes the film surrounding the droplet. The film is a multicomponent mixture formed by diffusion of surrounding gases. As a first approximation, properties in this film may be calculated by considering a mixture of 1/3 vaporized fuel and 2 / 3 surrounding mixture (see Ref. 33 for more details).

LAMINAR COUNTERFLOW SPRAY DIFFUSION FLAMES Let us now assume that the droplets are sufficiently small so that the drag may be expressed in terms of the Stokes law. In this case, the vector drag f is given in terms of vector velocities Vg and Vt by: f = 3"n"d //xg(Ve - vl). Equations 1-9 together with the transport relations, thermodynamic properties, gaseous state law, flux relations, and transfer term expressions constitute a closed system. Finally, for the gas-phase boundary conditions one specifies the temperatures, the mass fractions, and the velocities at the two nozzle exhaust sections. For the liquid phase the boundary conditions are only imposed at the lower nozzle exit. One specifies the temperature, initial droplet diameter, droplet number density, and initial velocity. NUMERICAL METHOD

Two numerical methods have been devised to solve the previous system of differential equations. In the first method, a small artificial viscosity is added to the liquid phase equations. This is needed because these expressions do not contain viscous terms. It has been verified that the artificial viscosity did not affect the solution. The set of equations for the gaseous and liquid phases is then replaced by a fully coupled set of discrete relations. The solution of this system is then based on a global adaptive nonlinear method using Newton iterations (see Refs. 34 and 35). The basic solution procedure is that developed by Smooke [36]. As the flame front and the droplet vaporization front (i.e., where the droplet diameter vanishes) are generally separated by only a few millimeters (1-2 mm), an adaptive gridding procedure places many points in the vicinity of these two fronts. As a consequence, the liquid phase is calculated even at points where there are no droplets and the gas phase is calculated on a grid which is too fine in the neighborhood of the vaporization front. Thus, to reduce the calculation time, a second method was devised to solve these equations. The two phases are treated separately. The gaseous phase is solved by the standard Newton iteration technique as

269

in the first method. Using the results from the gaseous phase calculation, we perform a marching solution of the liquid-phase equations based on a simple explicit integration method (without adding artificial viscosity). This liquid-phase solution is then used to determine the gas-phase source terms and a new global iteration is started to solve the gaseous phase. Generally four to five global iterations suffice to converge to the final solution. The gain in CPU time is about 20% compared with the first method. Thermochemical quantities and transport coefficients in both methods are evaluated using vectorized versions of CHEMKIN [28] and TRANSPORT [29] packages (for vectorization see Ref. 37). NUMERICAL RESULTS Let us study the structure of a counterflow involving an air stream on one side and on the other side a spray of n-heptane convected by a nitrogen stream containing a small amount of propane. The reaction mechanism for combustion of n-heptane involving 38 species consists of two global parts. The first part describes the decomposition of n-heptane into smaller hydrocarbons and involves 28 reactions (Table 1). These reactions originate from a recent work of Peters [38], who simplifies the detailed mechanism proposed in Ref. 39. Reactions 920 in Table l have been added after discussions with Professor Warnatz. The second part of the kinetic scheme is a detailed mechanism covering the combustion of hydrocarbons derived from C3H 8. It involves 231 direct reactions [40]. At this point, one may question if it is necessary to use detailed kinetics with many elementary reactions and complex thermodynamic and transport models while the droplet evaporation mechanism and the interphase transfer terms are much less refined. This is only apparently inconsistent. Indeed, a detailed kinetic scheme is required if one wishes to describe critical conditions like extinction because the flame is located in the gaseous phase. The evaporation model should at least allow a description of the vaporization kinetics but it may be more approximate because the ratio of the latent heat

270

N. D A R A B I H A E T AL. TABLE 1 n-Heptane Decomposition Mechanism due to Peters and Warnatz a Reaction 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12, 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

NC7H16 + H --* 1C7H15 + H2 NC7H16 + H ~ 2C7H15 + H 2 NCTH16 + H ~ 3C7H15 + H z NC7HI6 + H ~ 4C7H15 + H2 NC7H16 + OH ~ IC7HI5 + H 2 0 NC7HI6 + OH ~ 2C7H1s + H 2 0 NC7H16 + OH ~ 3C7H15 + H 2 0 NC7H16 + OH ~ 4C7Hls + H 2 0 NC7HI6 + O ---, 1C7H15 + O H NC7H16 + O ---, 2C7Ht5 + OH NCTH16 + O ~ 3C7H15 + OH NCTHI6 + O --+ 4C7H15 + O H NCTHI6 + HO 2 ~ 1C7H15 + H 2 0 2 NC7H16 + HO 2 ~ 2C7H15 + H 2 0 2 NCTH16 + HO E --~ 3C7H15 + H 2 0 2 NC7HI6 + HO 2 --+ 4C7H15 + H 2 0 2 NCvH16 + CH 3 ~ 1C7H15 + CH 4 NC7H|6 + CH 3 --~ 2C7H15 + CH 4 NCTHI6 + CH 3 --~ 3C7H15 + CH 4 NC7HI6 + CH 3 ~ 4C7H15 + CH 4 NC7HI6 --* C s H H + C2H 5 NC7HI6 ~ C4H 9 + C3H 7 1C7H15 -.o C2H4 + C2H4 + C3H7 2C7H15 --~ C3H 6 + C4H 9 3C7H15 --q.C3H 6 + C4H 9 4C7H15 ~ C3H 6 + C4H 9 C s H H ~ C2H 4 + C3H 7 C4H 9 ~ C2H 5 + C2H 4

A

/3

E

1.300E14 2.000E14 2.000E14 1.000El4 6.600E06 2.500E08 2.500E08 1.200E08 2.300E06 6.400E05 6.400E05 3.200E05 1.120E13 6.800E12 6.800E12 3.420E12 3.000E12 1.600E12 1.600E12 8.000Ell 2.000E16 2.000E16 5.000E13 5.000E13 5.000E13 5.000E13 5.000E13 2.500E13

0.0 0.0 0.0 0.0 2.0 1.46 1.46 1.46 2.4 2.5 2.5 2.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9700. 8340. 8340. 8340. 645. -190. -190. -190. 1590. 5000. 5000. 5000. 19400. 17000. 17000. 17000. 11600. 9500. 9500. 9500. 81260. 81260. 28680. 28680. 28680. 28680. 28680. 28820.

a These reactions are completed by a detailed mechanism due to Warnatz for the combustion of C3H 8 and lower hydrocarbons. Rate coefficients are in the form k: = ATOexp(-E/RT). Units are moles, cubic centimeters, seconds, Kelvins, and cal/mol.

to the heat release of the gaseous reaction is much smaller than unity. The position of the vaporization front will be correctly obtained even with a rough evaporation model. In these calculations the distance between the two injectors (Fig. 1) is equal to 2 cm, both streams are at an initial temperature of 300 K and the pressure is equal to one atmosphere. The droplet velocity matches the gas velocity at the injector exit and it is equal to 84 cm s -~ . The air stream velocity at the injector face is equal to 74 cm s -1. The initial droplet diameter is d t - - 2 2 /~m and the droplet number density is n = 30000 droplets per cm -3. The liquid mass flow rate per unit area (mass flux) is in this case 0.0099 g cm -2 s - l , the diluted propane mass fraction is Yc3H8 = 0.16 and its total (C3H 8 + N 2) mass flow rate per unit area is 0.101 g cm -2 s -1 . The air mass flow rate per

unit area delivered by the other nozzle is 0.089 g cm -2 s - l . Figures 9-16 represent the structure of this spray counterflow flame. The profile of the gas-phase temperature Tg is similar to that of a gaseous counterflow diffusion flame (Fig. 9). A closer view (Fig. 10) indicates that the temperature Tg decreases slightly because the droplets vaporize. It then increases at about 8 mm from the left injector and reaches its maximum value of 2200 K at about 11 mm from the left injector (Fig. 9). The temperature then decreases on the air side and is equal to the ambient temperature at about 13 mm from the left injector. The liquid phase temperature Tt decreases more rapidly than Tg, then starts to increase at about 8.3 mm, and reaches the vaporization temperature of 371.4 K at 9.2 mm from the left injector.

L A M I N A R C O U N T E R F L O W SPRAY D I F F U S I O N F L A M E S

¢0

271

1800.00

I,l~OoO

~-t

11101100

0.1!

C3H8

A ~coz

1.00

I l.fi0

1~0

y (~) Fig. 9. Calculated structure of a spray counterflow flame. The gas-phase temperature Tg is plotted as a function of the axial coordinate y. T h e spray of n - h e p t a n e is convected by a diluted propane stream (left side). See text for the calculation conditions.

The propane mass fraction remains constant until 8.3 mm from the left injector (Fig. 11), then decreases rapidly, and vanishes at 10 mm. The gaseous n-heptane mass fraction increases, due to the vaporization of the droplets, slightly from zero to 0.02 at 8.3 mm. It rapidly increases at the vaporization front and reaches its maximum value at 9.2 mm. It then decreases sharply by dissociation in hot gases and disappears at 10 mm. The CO 2 mass fraction follows the same type of variation as the gas temperature profile. The mass fraction profiles of some important minor species such as OH, H, and C H 3 are plotted in Fig. 12. O H appears in the hot region and nearly follows the same variation as the gas-phase temperature while H atoms are

y (~) Fig. 11. Major species mass fractions plotted as a function of the axial coordinate y. Conditions similar to those of Fig. 9.

present in a thinner region. C H 3 appears only on the fuel side of the flame front. The droplet diameter slightly decreases from its initial value of 22 tzm to approximately 20 /xm at 9 mm from the burner (Fig. 13), then decreases rapidly, and vanishes at 9.2 mm. A very good agreement is observed between the numerical calculations and the experimental results (shown as symbols in the figure). The distance between the flame front and the vaporization front is about 2 mm. This value is of the same order as the result obtained in Ref. 13 for the case of n-octane. It appears from Fig. 14 that the vaporization front is located in a zone where the temperature is less than 700 K. The gas-phase velocity v~ plotted in Fig. 15 has a zero slope at the injector exit planes. The absolute value of this velocity decreases and its

~

37O.OO

3311OO

310,O0

T1

2~0.C0

0.OO

I 0~o

I l J00

y (~ro) Fig. 10. Gas-phase temperature Tg and liquid phase temperature Tt are plotted as a function of the axial coordinate y. Conditions similar to those of Fig. 9.

l.ri0

I 0.~

OH

IJ0o

1~o

p.0o

y (~) Fig. 12. Some minor species mass fractions plotted as a function of the axial coordinate y. Conditions similar to those of Fig. 9.

272

N. D A R A B I H A ET AL. Num. OOExp. -

-

1~o

1500

10.00

P I

L~ -101)~0

y (an) Fig. 13. Droplet diameter plotted as a function of the axial coordinate y. Conditions similar to those of Fig. 9.

gradient is not constant throughout the flame. In this situation one cannot use the constant strain rate model. This point is now well documented in studies of flames formed by the counterflow of gaseous mixtures with that obtained in [10, 26, 27]. The stagnation point is located at an equal distance of 10 mm from each injector. A small acceleration of the gas is observed in the regions where the temperature begins to increase. The liquid phase velocity vt decreases more slowly then vg in the region located between the injector exit plane and the vaporization front. The liquid phase velocity vt reaches the same value as Cg at the back of the vaporization front. CONCLUDING REMARKS The structure of diffusion flames formed by the counterflow of a stream of air with a stream conveying a spray of n-heptane is investigated 1.00

r1 0~0

rg 0.40

0 ~ ° 0,00

/

\,

I

I

I

y (,~) Fig. 15. The gas phase normal velocity vg and the liquid phase normal velocity vt plotted as a function of the axial coordinate y. Conditionssimilar to those of Fig. 9.

in this paper. The case of a spray of small droplets conveyed by a stream of nitrogen containing a small amount of propane is specifically considered. Experiments carried with tomographic imaging and particle size measurement systems indicate that a sharp vaporization front is established in the vicinity of the flame front. This configuration is retrieved numerically by considering the self-similar solutions of the balance equations describing the gas and liquid phases using detailed kinetics and transport. The gas-phase equations are first solved without mass source terms. Using this solution, the liquid-phase equations are then treated with a space marching scheme. This provides a first approximation of the mass source terms. These terms are then incorporated in the gasphase equations in order to start a new global iteration. A few iterations are necessary to converge to the solution. The flame structure determined in this way agrees well with the available experimental data. An important advantage of the present numerical scheme is that it utilizes the well-proven methods of calculation developed for gaseous flames. A parametric study of the flame response remains to be done in terms of the injection velocities, initial droplet diameter, initial droplet number density, initial temperatures, and pressure.

y (~) Fig. 14. Reduced droplet radius and reduced gas phase temperature plotted as a function of the axial coordinate y. Conditions similar to those of Fig. 9.

We wish to express our thanks to professor J. T. C. Liu and Dr. Simonin for helpful discussions o f the spray model and to Professor J. Warnatz for

LAMINAR

COUNTERFLOW

SPRAY

DIFFUSION

helpful discussions o f the reaction m e c h a n i s m . P. Versaevel helped us in carrying the calculations during his final year training period. P. Scouflaire provided considerable help in the experimental work. This work was supported by S E P within the P R C Moteurs Fusde and by the C o m m i s s i o n o f the European C o m m u n i t i e s within the J O U L E program.

24. 25. 26.

27. 28.

REFERENCES

29.

1. Williams, A., Combust. Flame 11:1-31 (1973). 2. Faeth, (3. M., Prog. Ener. Combust. Sci., 9:1-76 (1983). 3. Spalding, D. B., Combustion and Mass Transfer, Oxford, Pergamon, 1979. 4. Williams, F. A., Combustion Theory, Cummings, Menlo Park, CA, 1985. 5. Kuo, K. K., Principles of Combustion, Wiley, New York, 1986. 6. Labowsky, M., Combust. Sci. Technol. 22:217-226 (1980). 7. Bellan, J., and Harstad, K., Combust. Flame 79:272-286 (1990). 8. Tsuji, H., Prog. Eng. Combust. Sci. 8:93 (1982). 9. Seshadri, K., Puff, 1., and Peters, N., Combust. Flame 61:237-249 (1985). 10. Rolon, J. C., Veynante, D., Martin, J. P., and Durst, F., Exp. Fluids 11:313-324 (1991). 11. Chelliah, H. A., Law, C. K., Veda, T., Smooke, M. D., and Williams, F. A., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp. 503-511. 12. (3reenberg, J. B., AIbagli, D., and Tambour, Y, Cornbust. Sci. Technol. 50:217-226 (1986). 13. Continillo, (3., and Sirignano, W. A., Combust. Flame 81:325-340 (1990). 14. Graves, D. B., and Wendt, J. O. L., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, pp. 1189-1196. 15. Wendt, J. O. L., Kram, B. M., MasteUer, M. M., and McCaslin, B. D., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, pp. 419-426. 16. Chen, Z. H., Lin, T. H., and Sohrab, S. H., Combust. Sci. Technol. 60:63-77 (1988). 17. Rolon, J. C., Ph.D. thesis, Ecole Centrale Paris, 1988. 18. Djavdan, E., Ph.D. thesis, Ecole Centrale Paris, 1990. 19. Lobdell, D. D., J. Acoust. Soc. Am. 43:229-231 (1968). 20. Bohren, G. F., and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983. 21. Mizutani, Y., Uga, Y., and Mishimoto, T., Bull. JSME 13:83, 620-627 (1972). 22. Eisner, E., J. Acoust. Soc. Am. 35:9 (1963). 23. Neppiras, E., Ultrasonics Int. Conf. Proc., 295-302, 1973.

30.

31.

32.

33.

34. 35. 36. 37.

FLAMES

273

Kouzelis, D., Candel, S. M., Esposito, E., and Zikikout, S., Panicle Charact. 4:151-156 (1987). Tardieu, A., Candel, S. M., and Esposito, E., A/AA Prog. Astronaut. Aeronaut. 95:736-749 (1984). Kee, R. J., Miller, J. A., Evans, G. H., and DixonLewis, G., Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 1479-1494. Smooke, M. D., Crump, J., Seshadri, K., and Giovangigli, V., Yale University Report ME-100-90, 1990. Kee, R. J., Miller, J. A., and Jefferson, T. H., Sandia National Laboratories Report, SAND80-8003, 1980. Kee, R. J., Warnatz, J., and Miller, J. A., Sandia National Laboratories Report, SAND83-8209, 1983. Godsave, G. A. E., Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1953, pp. 818-830. Spalding, D. B., Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1953, pp. 847-864. Vargaftik, N. B., Handbook of Physical Properties of Liquids and Gases', Pure Substances and Mirtures, 2nd ed., Springer-Verlag, Berlin, 1987. Abramzon, B., and Sirignano, W. A., AIAA Paper 88-0636, 26th Aerospace Sciences Meeting, Reno, NV, January 1988. Giovangigli, V., and Smooke, M. D., Combust. Sci. Technol. 53:23 (1987). Darabiha, N., Giovangigli, V., Candel, S., and Smooke, M. D., Combust. Sci. Technol. 60:267-285 (1988). Smooke, M. D., J. Comp. Phys. 48:72-105 (1982). Darabiha, N., and Giovangigli, V., Proceedings"of In-

ternational Symposium on High Performance Computing (L. Delhaye and E. Gelenbe, Eds.), Elsevier, Amsterdam, 1989. Miiller, U. C., Peters, N., and Linan, A., Twenty-fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 777--784. 39. Westbrook, C. K., Warnatz, J., and Pitz, W. J.,

38.

Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 893-901. 40. Warnatz, J.. Private Communication.

Received 23 November 1991; revised 23 December 1992

APPENDIX: DERIVATION OF THE LIQUID PHASE EQUATIONS T h e l i q u i d p h a s e t r e a t e d as a c o n t i n u u m m a y b e d e s c r i b e d by t h e f o l l o w i n g c o n s e r v a t i o n equations:

Liquid phase Mass 1 d(xJnmtul) xj

dx

+

d ( n m t v l) dy

= -nrht,

(All

274

N. D A R A B I H A E T AL.

x momentum

Introducing these results in Eqs A1 to A5 one obtains:

1 d(xJnm,ul 2) xj

+

d(nmlv, Ut)

dx

Mass

dy

(A2)

= nfx - nrhtut,

d(nv l) d(m~) mr(1 + j)nUt + m t + nv t dy dy

y Momentum

= - nrh~.

1 d(xJnmlulV,) xj

+

x momentum

d(nmtvl 2)

dx

dy

= nmtg + nfy - nrhtv t,

(A3)

d(nv t) (1 + j)nmtUl 2 + mtUt------;~ + nmtUt 2 ay d(mlUl) + nv t - = nFx - nrhlUt.

Energy

= nr'ne( q - L ) - nrhtcptT~,

y momentum

(A4)

d(nv l) d(mtv l) (1 + j)nmtUlV t + m w t - + nv r ay dy = nmtg + nfy - nrhtv t.

Droplet number density

1 d(xJnul) dx

1 d(xJnmtut) dr

= o.

(AS)

= (1 + j ) n m l U ,,

1 d ( x i n m t u t 2) = x[(1 + j ) n m , U 7 + nm,Ut 2] x/ dx 1 d(x-inmtutvt) xj

dx

= (1 + j)nmtUlvt,

1 d ( x i n m l u l T t) = (1 +j)nmtUlTl, xI dx 1 d ( # n u l) x~

dx

(A8)

Energy

d(nv~)

+ - dy

The similarity analysis of the liquid phase (Ref. 13) leads to solutions of the form: Pt = Or(Y) = n(y)mt(y), u t = xUl(y), v t = vt(y), and Tl = Tt(y), where x and y denote radial and axial coordinates, respectively. Pl, nl, ml, Ul, Ul, and Tt denote liquid phase density, droplet number density, droplet mass, radial velocity, axial velocity, and temperature, respectively. Ut describes the y dependence of the axial ut.With these assumptions one obtains:

xj

(A7)

ay

1 d(xJnmlutT t) d(nmtvtTt) Cpl X j HX + cpl dy

xj

(A6)

Cpl(1 + j)nmtUtTt + CplmlTI

+ CplnUl

d(nv l) dy

d(mfl) dy

Droplet number density (1

+

j)nUt

dn v t +

- -

ay

(A10)

= 0,

where F x = f i x . NOw, by introducing the droplet number density equation (Eq. A10) in Eqs. A6-A9, the first two terms on the left side of these equations disappear and one finds: Mass d(mt) Ul dy

rh~.

(All)

x momentum d(mt) ay

d(Ui)

re,V, 2 + U , v , - - = - - - + mlUl--~y

= (1 + j ) n U t.

(A9)

= nrhl( q - L ) - nrhlC;lZI.

=

g

-

~h~.

(A12)

L A M I N A R C O U N T E R F L O W SPRAY DIFFUSION FLAMES

liquid phase to find the five unknowns: m t, Ut, vt, T', and n: Mass

y momentum v~ d ( m , ) -

-

dy

d(vt) +

mlv l

dy

275

= mtg

+ fy -

rhtv.

(A13)

dmt vt dy

th t .

(A16)

x momentum

Energy

mtVt 2 + mtvt--~y = Fx.

d ( m l) d(T') CplT'Vt----j77- + Cptmtv t ay

y momentum

uy

= rhl( q - L ) - rhtCptT'.

(A14)

Droplet number density dnv l (1 + j ) n U t + ~ = 0.

dy

(A17)

dvt mY1 dy = m l g + fy"

(A18)

Energy

aT,

(A15)

mlC plVt-~y = rht( q - L ).

(A19)

Droplet number density

Finally, by introducing the mass conservation Eq. A l l in Eqs. A12-A14 one can write the following system of 5 equations to describe the

(1 + j ) n U t +

dnv t

ay

= O.

(A20)